Coloring - Problem Solving

Determine the maximum number of kings that can be placed on a \(12\times12\) chessboard so that each king threatens exactly one other king.

Note: Each cell contains at most 1 king. Two kings threaten each other if they inhabit adjacent or diagonally adjacent cells.

The above figure of 45 squares can be covered completely with eleven L-tetrominos and one monomino. How many possible positions are there for the position of the monomino?

A grid with 3 rows and 52 columns is tiled with 78 identical \(2 \times 1\) dominoes. How many ways can this be done such that exactly two of the dominoes are vertical?

Details and Assumptions:

• The dominoes will cover the entire board. They are not allowed to jut over the board, or overlap with each other.

• Rotations and reflections (of the board) count as distinct ways.

• Convention: rows are horizontal and columns are vertical.

What is the minimum number of colors you need to color all sides of an icosahedron such that no two sides that join on an edge have the same color? (The one in the image wouldn't qualify since two faces of the same color share an edge.)

Note: It is okay if two sides meeting at a vertex have the same color, but not if they share an edge.